Chapter 7.4, Problem 53E

To determine

To find: The partial fraction decomposition of the given rational expression.

Answer to Problem 53E

The partial fraction decomposition of the given rational expression is $\frac{2}{x}-\frac{3}{x-1}+\frac{4}{x+1}$ .

Explanation of Solution

Given information:

The rational expression is $\frac{3x^2-7x-2}{x^3-x}$ .

Calculations:

The factored form of the denominator $x^3-x$ is $x(x-1)(x+1)$ .

Based on the factors of the denominator, the decomposition form is as follows:

  $\frac{3x^2-7x-2}{x(x-1)(x+1)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}$

Now, multiply each side by the least common multiple, $x(x-1)(x+1)$ , to obtain the basic equation.

That is,

  $\begin{array}{l}3x^2-7x-2=A(x+1)(x-1)+Bx(x+1)+Cx(x-1)\\ \Rightarrow 3x^2-7x-2=A(x^2-1)+B(x^2+x)+C(x^2-x)\\ \Rightarrow 3x^2-7x-2=A{x}^2-A+B{x}^2+Bx+C{x}^2-Cx\\ \Rightarrow 3x^2-7x-2=(A+B+C)x^2+(B-C)x+(-A)\end{array}$

Comparing the coefficients of like terms, we get

  $\{\begin{array}{c}A+B+C=3\\ B-C=-7\\ -A=-2\end{array}$

Using $-A=-2$ , we get $A=2$ .

Substituting the value in the first equation, we get

  $\begin{array}{l}A+B+C=3\\ \Rightarrow 2+B+C=3\\ \Rightarrow B+C=1\end{array}$

Now, adding this equation to the second equation, we get

  $\begin{array}{l}(B+C)+(B-C)=1+(-7)\\ \Rightarrow 2B=-6\\ \Rightarrow B=-3\end{array}$

Using this,

  $\begin{array}{l}B+C=1\\ \Rightarrow -3+C=1\\ \Rightarrow C=4\end{array}$

Therefore, the partial fraction decomposition of the given rational expression is $\frac{2}{x}-\frac{3}{x-1}+\frac{4}{x+1}$ .

Conclusion:

Using Desmos graphing utility, we can observe that both curves are overlapping each other.